You have wildly complicated your circuit, assuming your symbols mean what the schematic indicates.
First, an instrumentation amp would look like
simulate this circuit – Schematic created using CircuitLab
You show the upper amplifier, which produces OutInv, as being connected to the same ground as R3 and R9. Under these circumstances, OutInv is forced to ground (plus the input offset of the op amp), so you could profitably get rid of the op amp. Furthermore, a gain of 100 is perfectly reasonable from a single op amp, so you could do the whole thing with 1 op amp, as George Herold suggested. A simple version would be
simulate this circuit
This will have a nominal gain of 101. Since an LM358 has a maximum offset voltage of 7 mV, you could have an offset error of 0.7 volts.
Let's assume, though, that the two input grounds are instead a separate ground, isolated from the output ground by some common-mode voltage. In this case, you have connected the upper op amp incorrectly. R1 should be connected to the - input of both amps. However, if you do this, you'll have gain of about 220, rather than 100. The equation for gain for an instrumentation amplifier (assuming R2 equals R3, R4 equals R5, and R6 = R7) is$$G=(1+\frac{2R_2}{R_1})\frac{R_6}{R_5}$$ Alternatively, depending on the output impedance of V1, you could simply use a single op amp set up as a difference amplifier with a gain of 100. You've shown the V1 as a voltage source, so this seems perfectly reasonable.
If you're worried about a stable output, I assume you're worried about noise from the input. This is best handled with an RC filter between V1 and the input.
Best Answer
There are 2 obvious possibilities for falling "gain" with increasing input level :
What you expect:
Vout = Vin * gain
And you are calculating gain as Gain = Vout/Vin
What you get:
Vout = (Vin + Voffset) * Gain.
So if Voffset = 5mv and gain = 5,
Vin = 5mv : Vout = (5mv + 5mv) * 5 = 50mv.
And you compute gain as Vout/Vin = 10.
Vin = 25mv : Vout = (25mv + 5mv) * 5 = 150mv
And you compute gain as Vout/Vin = 6.
This is a serious issue : it looks like you are attempting to amplify microvolts of DC with a device with millivolts of input offset voltage. This offset voltage is an error signal that will swamp the signal of interest.
What bandwidth do you need, and have you looked at chopper stabilised opamps to eliminate DC offset?
One opamp that may be useful to you is the ADA4528 which is further described in this Q&A which claims an offset voltage of only 2.5 uV.